Meaning
The banded tensor-selector class is the degree-two class obtained by evaluating the band’s degree-three class along the certified selected loops.
The selector-comparison certificate supplies a map ℓ from the selected filling space to the loop space of the coefficient band. Loop transgression lowers degree by one, and pullback gives s=ℓ∗Tr_S¹(b_D). Both operations and the selected loop data are part of the definition.
Example (illustrative)
If b_D=0, its transgression and pullback are zero. A nonzero b_D can also have zero pullback on a particular selected filling space, because that selection may not detect the whole class.
Scope and limits
This class is not automatically the relative filling class r_D. They lie in different cohomology groups; relating them requires a specified comparison theorem.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 11.4.